Beautiful Pairs of Algebraically Closed Valued Fields and Non-Standard Frobenius

Explore the theory of beautiful pairs for algebraically closed valued fields (ACVF) in this advanced mathematical lecture that generalizes Poizat's work from stable to unstable theories. Discover how beautiful pairs provide a semantic approach to definability questions in ACVF, particularly regarding important spaces of definable types. Learn about the establishment of strict pro-definability for all definable types and bounded definable types on algebraic varieties defined over valued fields, which serve as definable analogues of the Zariski-Riemann space and Huber space. Understand the main theoretical framework through an Ax-Kochen-Ershov type reduction approach. Examine ongoing research developments involving non-standard Frobenius automorphisms added to ACVF structures, including the surprisingly simple axioms that emerge when these enhanced structures also prove to be beautiful pairs.